Question

Write the number of real roots of the equation \[(x - 1 )^2 + (x - 2 )^2 + (x - 3 )^2 = 0\].

Answer 1

\[(x - 1 )^2 + (x - 2 )^2 + (x - 3 )^2 = 0\]

\[ \Rightarrow x^2 + 1 - 2x + x^2 + 4 - 4x + x^2 + 9 - 6x = 0\]

\[ \Rightarrow 3 x^2 - 12 x + 14 = 0\]

Comparing the given equation with the general form of the quadratic equation 

\[a x^2 + bx + c = 0\] we get 

\[a = 3 , b = - 12\] and \[c = 14 .\]

\[D = b^2 - 4ac = \left( - 12 \right)^2 - 4 \times 3 \times 14 = 144 - 168 = - 24\]

\[\text { Since the value of D is less than 0, the given equation has no real roots } .\]